For smooth welfare functions of user utilities and item exposures, the update direction (REF ) can be computed with only one top-\(K\)  sorting operation per user [1]}.
In our case, the update is given by the following result, where \(\mathop {\text{top-}K}({z}) = \lbrace  \sigma \in \mathfrak {S}_n: z_{\sigma (1)}\ge \ldots \ge z_{\sigma (K)} \text{ and } \forall k\ge K, z_{\sigma (K)} \ge z_{\sigma (k)}\rbrace \) is the set of permutations that sort the \(k\)  largest elements in \({z}\) .
